Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

Bhatt, Ashish; Floyd, Dwayne; Moore, Brian E.

doi:10.1007/s10915-015-0062-z

Cited by 31 publications

(36 citation statements)

References 16 publications

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“…However, the scheme does preserve such conformal conservation laws in special cases, as is demonstrated in the following section. It is also important to notice that the external forcing plays no role in the variational equation, so any multi-conformal-symplectic scheme will also preserve the conservation law (2) in the presence of external forcing.…”

Section: Symplectic Euler Spatial Discretizationmentioning

confidence: 99%

“…Then, the ideas were extended to general linearly damped multisymplectic PDEs, with application to damped semi-linear waves equations and damped NLS equations, where midpoint-type discretizations (similar to the Preissmann box scheme and midpoint discrete gradient methods) were shown to preserve various dissipative properties of the governing equations [28]. Other similar methods have been developed based on an evenodd Strang splitting for a modfied (non-diffusive) Burger's equation [2], the average vector field method for a damped NLS equation [11], and an explicit fourth-order Nyström method with composition techniques for damped acoustic wave equations [5].…”

Section: Introductionmentioning

confidence: 99%

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Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces

Bhatt

Moore

2019

Journal of Computational and Applied Mathematics

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Structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as Korteweg-de Vries, Klein-Gordon, Schrödinger, and Camassa-Holm equations, all with damping/driving terms and time-dependent coefficients. Since key features of the PDEs under consideration are described by local conservation laws, which are independent of the boundary conditions, the proposed (second-order in time) discretizations are developed with the intent of preserving those local conservation laws. The methods are respectively applied to a damped-driven nonlinear Schrödinger equation and a damped Camassa-Holm equation. Numerical experiments illustrate the structure-preserving properties of the methods, as well as favorable results over other competitive schemes.

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Section: Symplectic Euler Spatial Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces

Bhatt

Moore

2019

Journal of Computational and Applied Mathematics

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“…Although this is a challenging problem, there exist a few results on analyses on this topic (e.g. [6]). The results of these analyses could give an insight on the qualitative acoustical analyses of computations of musical sounds.…”

Section: Resultsmentioning

confidence: 99%

Geometric-integration tools for the simulation of musical sounds

Ishikawa

Michels

Yaguchi

2018

Japan J. Indust. Appl. Math.

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During the last decade, much attention has been given to sound rendering and the simulation of acoustic phenomena by solving appropriate models described by Hamiltonian partial differential equations. In this contribution, we introduce a procedure to develop appropriate tools inspired from geometric integration in order to simulate musical sounds. Geometric integrators are numerical integrators of excellent quality that are designed exclusively for Hamiltonian ordinary differential equations. The introduced procedure is a combination of two techniques in geometric integration: the semi-discretization method by Celledoni et al. (J Comput Phys 231:6770-6789, 2012) and symplectic partitioned Runge-Kutta methods. This combination turns out to be a right procedure that derives numerical schemes that are effective and suitable for computation of musical sounds. By using this procedure we derive a series of explicit integration algorithms for a simple model describing piano sounds as a representative example for virtual instruments. We demonstrate the advantage of the numerical methods by evaluating a variety of numerical test cases.

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“…We point out that if the prefactor in front of dq ∧ dp is initially not in an exponential form, we can always rewrite it into an exponential form as long as it is a constant value between zero and one. Following [5,20], we can show that the YBABY method (29) is conformal symplectic: dq n+1 ∧ dp n+1 = e −γh/2 dq n+1 ∧ dp n+2/4 , = e −γh/2 dq n ∧ dp n+2/4 , = e −γh dq n ∧ dp n .…”

Section: The Ybaby Methodsmentioning

confidence: 99%

Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

Shang

Öttinger

2020

Proc. R. Soc. A.

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We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

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Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

Cited by 31 publications

References 16 publications

Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces

Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces

Geometric-integration tools for the simulation of musical sounds

Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

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